$L(t)$ models the length of each day (in minutes) in Manila, Philippines $t$ days after the spring equinox. Here, $t$ is entered in radians. $L(t) = {52}\sin\left({\dfrac{2\pi}{365}}t\right) + {728}$ What is the first day after the spring equinox that the day length is $750$ minutes? Round your final answer to the nearest whole day.
Explanation: Converting the problem into mathematical terms $L(t) = {52}\sin\left({{\dfrac{2\pi}{365}}}t\right) + {728}$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{365}}}=365$ days. We want to find the first solution to the equation $L(t)=750$ within the period $0<t<365$. The answer The equation's two solutions within the desired period (rounded to the nearest whole day) are $25$ and $157$. Therefore, the first $750$ -minute day is $25$ days after the spring equinox.